What is the Area Not Shared by the Circles?

Date: 3 Mar 1995 10:01:35 -0500
From: Mary Basse
Subject: Geometry
When I was taking the SAT test, I ran across this problem. Will
you please help me? Two circles intersect such that their centers
and their points of intersection form a square with each side
equal to 3. What is the total area of the sections of the square
that are not shared by both circles?
Thank you.
Russell
mbasse@isbe.state.il.us

Date: 3 Mar 1995 10:41:53 -0500
From: Dr. Ken
Subject: Re: Geometry
Hello there!
This is one of the nicer problems I've seen on the SAT (I'm ordinarily
not a fan of their math problems, or even their answers; they have
been known to use incorrect answers as correct). Here's a hint on
how you might approach it. You know that the area of the square must
be 9, right? So what we want to do is subtract the football-shaped
region in the middle.
So how do we find its area? Well, here's what I'd do. You can find the
area of the part of the square that makes up 1/4 of one of the circles,
right? If you add two of these regions together, you will get the area
of the square plus some overlap. This overlap is the area of the
football. And once you've found the area of the football, you can
subtract it from 9 to find out the answer to the problem.
Thanks for the question, and let us know if you need some more help!
-Ken "Dr." Math